I will explain a novel construction of K3 surfaces as moduli spaces of singular equivariant instantons on a 4-torus. This gives the first hyper-Kahler quotient construction of a compact non-toroidal manifold and yields explicit formulae for K3 metrics near torus orbifold limits. I will also describe a novel Fredholm theory — and its development using interesting ideas from microlocal analysis — for Laplacians and Dirac operators constructed out of these very singular connections which enables the construction. I will next introduce a variant of the Donaldson-Uhlenbeck-Yau theorem that operates in this setting, some of its consequences (such as non-emptiness of our moduli spaces), and the novel notion of stability for these singular connections that enters into the theorem. This theorem is proved by studying the gradient flow equation for the Yang-Mills functional, and I will describe the Fredholm theory for the heat operator that undergirds the proof of the short- and long-time existence and regularity of the flow. While these results are all proved for singular connections on a 4-torus, they are expected to generalize to enable the study of vast new classes of gauge-theoretic moduli spaces consisting of objects with severe singularities in codimension at least three. Based on joint work with Arnav Tripathy and Andras Vasy.